metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.171D6, C6.342- 1+4, C4⋊Q8⋊9S3, C4.37(S3×D4), C4⋊C4.122D6, (C4×S3).13D4, D6.48(C2×D4), C12.69(C2×D4), D6⋊Q8⋊47C2, (C2×Q8).167D6, C6.98(C22×D4), C42⋊2S3⋊25C2, C42⋊7S3⋊26C2, D6.D4⋊45C2, (C2×C6).268C24, D6⋊C4.49C22, Dic3.53(C2×D4), Dic3⋊Q8⋊26C2, C12.23D4⋊25C2, (C2×C12).101C23, (C4×C12).209C22, (C6×Q8).135C22, (C2×D12).171C22, C22.289(S3×C23), Dic3⋊C4.165C22, (C22×S3).230C23, C2.35(Q8.15D6), C3⋊5(C23.38C23), (C2×Dic6).188C22, (C4×Dic3).159C22, (C2×Dic3).140C23, (C2×S3×Q8)⋊12C2, C2.71(C2×S3×D4), (C3×C4⋊Q8)⋊10C2, (S3×C2×C4).142C22, (C2×Q8⋊3S3).7C2, (C3×C4⋊C4).211C22, (C2×C4).217(C22×S3), SmallGroup(192,1283)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.171D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c5 >
Subgroups: 688 in 270 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22×S3, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C4⋊Q8, C22×Q8, C2×C4○D4, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, C3×C4⋊C4, C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, S3×Q8, Q8⋊3S3, C6×Q8, C23.38C23, C42⋊2S3, C42⋊7S3, D6.D4, D6⋊Q8, Dic3⋊Q8, C12.23D4, C3×C4⋊Q8, C2×S3×Q8, C2×Q8⋊3S3, C42.171D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, 2- 1+4, S3×D4, S3×C23, C23.38C23, C2×S3×D4, Q8.15D6, C42.171D6
►On 96 points
(1 52 7 58)(2 59 8 53)(3 54 9 60)(4 49 10 55)(5 56 11 50)(6 51 12 57)(13 26 19 32)(14 33 20 27)(15 28 21 34)(16 35 22 29)(17 30 23 36)(18 25 24 31)(37 64 43 70)(38 71 44 65)(39 66 45 72)(40 61 46 67)(41 68 47 62)(42 63 48 69)(73 96 79 90)(74 91 80 85)(75 86 81 92)(76 93 82 87)(77 88 83 94)(78 95 84 89)
(1 82 33 46)(2 47 34 83)(3 84 35 48)(4 37 36 73)(5 74 25 38)(6 39 26 75)(7 76 27 40)(8 41 28 77)(9 78 29 42)(10 43 30 79)(11 80 31 44)(12 45 32 81)(13 92 57 72)(14 61 58 93)(15 94 59 62)(16 63 60 95)(17 96 49 64)(18 65 50 85)(19 86 51 66)(20 67 52 87)(21 88 53 68)(22 69 54 89)(23 90 55 70)(24 71 56 91)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 42 7 48)(2 47 8 41)(3 40 9 46)(4 45 10 39)(5 38 11 44)(6 43 12 37)(13 90 19 96)(14 95 20 89)(15 88 21 94)(16 93 22 87)(17 86 23 92)(18 91 24 85)(25 74 31 80)(26 79 32 73)(27 84 33 78)(28 77 34 83)(29 82 35 76)(30 75 36 81)(49 66 55 72)(50 71 56 65)(51 64 57 70)(52 69 58 63)(53 62 59 68)(54 67 60 61)
G:=sub<Sym(96)| (1,52,7,58)(2,59,8,53)(3,54,9,60)(4,49,10,55)(5,56,11,50)(6,51,12,57)(13,26,19,32)(14,33,20,27)(15,28,21,34)(16,35,22,29)(17,30,23,36)(18,25,24,31)(37,64,43,70)(38,71,44,65)(39,66,45,72)(40,61,46,67)(41,68,47,62)(42,63,48,69)(73,96,79,90)(74,91,80,85)(75,86,81,92)(76,93,82,87)(77,88,83,94)(78,95,84,89), (1,82,33,46)(2,47,34,83)(3,84,35,48)(4,37,36,73)(5,74,25,38)(6,39,26,75)(7,76,27,40)(8,41,28,77)(9,78,29,42)(10,43,30,79)(11,80,31,44)(12,45,32,81)(13,92,57,72)(14,61,58,93)(15,94,59,62)(16,63,60,95)(17,96,49,64)(18,65,50,85)(19,86,51,66)(20,67,52,87)(21,88,53,68)(22,69,54,89)(23,90,55,70)(24,71,56,91), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,42,7,48)(2,47,8,41)(3,40,9,46)(4,45,10,39)(5,38,11,44)(6,43,12,37)(13,90,19,96)(14,95,20,89)(15,88,21,94)(16,93,22,87)(17,86,23,92)(18,91,24,85)(25,74,31,80)(26,79,32,73)(27,84,33,78)(28,77,34,83)(29,82,35,76)(30,75,36,81)(49,66,55,72)(50,71,56,65)(51,64,57,70)(52,69,58,63)(53,62,59,68)(54,67,60,61)>;
G:=Group( (1,52,7,58)(2,59,8,53)(3,54,9,60)(4,49,10,55)(5,56,11,50)(6,51,12,57)(13,26,19,32)(14,33,20,27)(15,28,21,34)(16,35,22,29)(17,30,23,36)(18,25,24,31)(37,64,43,70)(38,71,44,65)(39,66,45,72)(40,61,46,67)(41,68,47,62)(42,63,48,69)(73,96,79,90)(74,91,80,85)(75,86,81,92)(76,93,82,87)(77,88,83,94)(78,95,84,89), (1,82,33,46)(2,47,34,83)(3,84,35,48)(4,37,36,73)(5,74,25,38)(6,39,26,75)(7,76,27,40)(8,41,28,77)(9,78,29,42)(10,43,30,79)(11,80,31,44)(12,45,32,81)(13,92,57,72)(14,61,58,93)(15,94,59,62)(16,63,60,95)(17,96,49,64)(18,65,50,85)(19,86,51,66)(20,67,52,87)(21,88,53,68)(22,69,54,89)(23,90,55,70)(24,71,56,91), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,42,7,48)(2,47,8,41)(3,40,9,46)(4,45,10,39)(5,38,11,44)(6,43,12,37)(13,90,19,96)(14,95,20,89)(15,88,21,94)(16,93,22,87)(17,86,23,92)(18,91,24,85)(25,74,31,80)(26,79,32,73)(27,84,33,78)(28,77,34,83)(29,82,35,76)(30,75,36,81)(49,66,55,72)(50,71,56,65)(51,64,57,70)(52,69,58,63)(53,62,59,68)(54,67,60,61) );
G=PermutationGroup([[(1,52,7,58),(2,59,8,53),(3,54,9,60),(4,49,10,55),(5,56,11,50),(6,51,12,57),(13,26,19,32),(14,33,20,27),(15,28,21,34),(16,35,22,29),(17,30,23,36),(18,25,24,31),(37,64,43,70),(38,71,44,65),(39,66,45,72),(40,61,46,67),(41,68,47,62),(42,63,48,69),(73,96,79,90),(74,91,80,85),(75,86,81,92),(76,93,82,87),(77,88,83,94),(78,95,84,89)], [(1,82,33,46),(2,47,34,83),(3,84,35,48),(4,37,36,73),(5,74,25,38),(6,39,26,75),(7,76,27,40),(8,41,28,77),(9,78,29,42),(10,43,30,79),(11,80,31,44),(12,45,32,81),(13,92,57,72),(14,61,58,93),(15,94,59,62),(16,63,60,95),(17,96,49,64),(18,65,50,85),(19,86,51,66),(20,67,52,87),(21,88,53,68),(22,69,54,89),(23,90,55,70),(24,71,56,91)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,42,7,48),(2,47,8,41),(3,40,9,46),(4,45,10,39),(5,38,11,44),(6,43,12,37),(13,90,19,96),(14,95,20,89),(15,88,21,94),(16,93,22,87),(17,86,23,92),(18,91,24,85),(25,74,31,80),(26,79,32,73),(27,84,33,78),(28,77,34,83),(29,82,35,76),(30,75,36,81),(49,66,55,72),(50,71,56,65),(51,64,57,70),(52,69,58,63),(53,62,59,68),(54,67,60,61)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | 2- 1+4 | S3×D4 | Q8.15D6 |
| kernel | C42.171D6 | C42⋊2S3 | C42⋊7S3 | D6.D4 | D6⋊Q8 | Dic3⋊Q8 | C12.23D4 | C3×C4⋊Q8 | C2×S3×Q8 | C2×Q8⋊3S3 | C4⋊Q8 | C4×S3 | C42 | C4⋊C4 | C2×Q8 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 2 | 2 | 4 |
Matrix representation of C42.171D6 ►in GL8(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 4 | 8 | 6 |
| 0 | 0 | 0 | 0 | 9 | 3 | 6 | 5 |
| 0 | 0 | 0 | 0 | 4 | 0 | 10 | 4 |
| 0 | 0 | 0 | 0 | 0 | 9 | 9 | 10 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 11 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 11 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| 12 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,3,9,4,0,0,0,0,0,4,3,0,9,0,0,0,0,8,6,10,9,0,0,0,0,6,5,4,10],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0],[0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,12,0,0,1,0,0,0,0,11,0,0,1,0,0,0,0,0,11,1,0],[12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0] >;
C42.171D6 in GAP, Magma, Sage, TeX
C_4^2._{171}D_6 % in TeX
G:=Group("C4^2.171D6"); // GroupNames label
G:=SmallGroup(192,1283);
// by ID
G=gap.SmallGroup(192,1283);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,100,675,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^5>;
// generators/relations